Vol. 6, No. 7, 2012

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Grothendieck's trace map for arithmetic surfaces via residues and higher adèles

Matthew Morrow

Vol. 6 (2012), No. 7, 1503–1536
Abstract

We establish the reciprocity law along a vertical curve for residues of differential forms on arithmetic surfaces, and describe Grothendieck’s trace map of the surface as a sum of residues. Points at infinity are then incorporated into the theory and the reciprocity law is extended to all curves on the surface. Applications to adelic duality for the arithmetic surface are discussed.

Keywords
reciprocity laws, higher adèles, arithmetic surfaces, Grothendieck duality, residues
Mathematical Subject Classification 2010
Primary: 14H25
Secondary: 14B15, 14F10
Milestones
Received: 26 June 2011
Revised: 25 October 2011
Accepted: 13 December 2011
Published: 4 December 2012
Authors
Matthew Morrow
Department of Mathematics
University of Chicago
5734 University Avenue
Chicago, IL 60637-1546
United States
http://math.uchicago.edu/~mmorrow/